A function is named continuos if for every two values assigned to the variable in the function, if those two are not too different, then outputs of both them are similar too. In other words, a function is named continuos if for every number X in the domain, the outputs of all values close to X are close to the X output too. It is common to hear that a continuos function is one that can be drawn in a paper without lifting the pencil. Most of usual fuctions are continuos: polynomial functions, trygonometrical functions and their inverse functions, exponential and logarithmic functions, rational functions, are all of the continuos, and there are many others. Continuos functions domains are intervals. Intervals are collections of numbers that for every pair in the collection, every real number between them two is also in the collection. So, functions can only be continuos if the domain are real numbers; otherwhise, pencil would have to be lifted in some points when drawing it. Functions can be continuos on specific points, or in a complete interval, when it is continuos in all the points in the interval. Continuos functions have some interesting characteristics. One of them is that if a function is differentiable in a point, then it must be continuos in that same point. Other characteristic is that there is always a primitive for a continuos function. Primitives are other functions associated to the original one that, when differentiated, results into the given. Other important think that might be mentioned is that adding two continuos functions results into a third one that must be continuos too, and so does multiplying two continuos functions.
Difference between continuos functions and those that are not continuos, which are called discontinous functions, was enunciated in Bolzano’s Theorem, which postulated that if a function is continuos on an interval from a to b, when you calculate the output of a and b, then, all the possible values between f(a) and f(b) are outputs of the number inside the interval.
There is no classification for continuos functions, but you can compare some characteristics between some of them, like trygonometrical, polynomials, exponential and logarithmic functions.
Fuctions
Characteristics.
Trygonometrical functions
Inverse trigonometrical functions
Polynomial Functions
Exponential Function
Logarithmic Function
On which intervals are they defined?
Defined for every real number.
Defined on the interval from -1 to 1.
Defined for every real number
Defined for every real number
Defined for every possitive real number
Which values do they take?
They take values between -1 and 1
They take values between 0 and 2∏
They take values on every real number.
It takes values on every real number.
It takes values up to -1 and over.
Do they take a minimun/ maximun value?
Yes, they can take a maximun and a minimun value.
Yes, they take maximun and minimun values.
They can take max/ min values or not, depending on the polynomial.
It take a min or max value depending on the variable (if it is possitive or no)
It takes a minimun value, but does not have a maximun one.
So,too coloquial... Use: Therefore, when talking about continuos functions we know:
Common characteristics: they are defined on intervals of real numbers, they take values on intervals too, they are differentiable, and there is always a primitive for them.
Particular Characteristics: :they can or can not have maximal or minimal values, they can be defined over small intervals or for every real number, and sometimes there is an inverse function and sometimes there is not.
Super
Continuos Functions.
A function is named continuos if for every two values assigned to the variable in the function, if those two are not too different, then outputs of both them are similar too. In other words, a function is named continuos if for every number X in the domain, the outputs of all values close to X are close to the X output too. It is common to hear that a continuos function is one that can be drawn in a paper without lifting the pencil. Most of usual fuctions are continuos: polynomial functions, trygonometrical functions and their inverse functions, exponential and logarithmic functions, rational functions, are all of the continuos, and there are many others.
Continuos functions domains are intervals. Intervals are collections of numbers that for every pair in the collection, every real number between them two is also in the collection. So, functions can only be continuos if the domain are real numbers; otherwhise, pencil would have to be lifted in some points when drawing it. Functions can be continuos on specific points, or in a complete interval, when it is continuos in all the points in the interval.
Continuos functions have some interesting characteristics. One of them is that if a function is differentiable in a point, then it must be continuos in that same point. Other characteristic is that there is always a primitive for a continuos function. Primitives are other functions associated to the original one that, when differentiated, results into the given. Other important think that might be mentioned is that adding two continuos functions results into a third one that must be continuos too, and so does multiplying two continuos functions.
Difference between continuos functions and those that are not continuos, which are called discontinous functions, was enunciated in Bolzano’s Theorem, which postulated that if a function is continuos on an interval from a to b, when you calculate the output of a and b, then, all the possible values between f(a) and f(b) are outputs of the number inside the interval.
There is no classification for continuos functions, but you can compare some characteristics between some of them, like trygonometrical, polynomials, exponential and logarithmic functions.
Characteristics.
functions
functions
Functions
Function
Function
-1 to 1.
Common characteristics: they are defined on intervals of real numbers, they take values on intervals too, they are differentiable, and there is always a primitive for them.
Particular Characteristics: :they can or can not have maximal or minimal values, they can be defined over small intervals or for every real number, and sometimes there is an inverse function and sometimes there is not.